A distance relay is a protection relay, which is used to protect electric systems and parts thereof, such as electric lines, against faults. A traditional method of generating a distance protection function with circular characteristics is to compare the angle between two voltage phasors: an operating voltage phasor S1 and a polarizing voltage phasor S2. Phasor S1 is also commonly known as “the line drop compensated voltage”. They take the following form:S1=ULx+dir·(ILx+IN·KN)·Z1set  (eq 2.1)S2=ULxpola  (eq 2.2)                where        ULx=voltage phasor of faulted phase(s) x. For phase-to-earth elements the voltage is UL1, UL2 or UL3. For phase-to-phase elements the voltage is UL12, UL23 or UL31.        dir=−1, if operation direction is forward, +1, if operation direction is reverse.        ILx=current phasor of faulted phase(s). For phase-to-earth elements the current is IL1, IL2 or IL3. For phase-to-phase elements the current is IL12, IL23 or IL31.        IN=residual current (IL1+IL2+IL3) phasor. This term is zero for phase-to-phase elements.        KN=residual compensation factor=(Z0set−Z1set)/(3·Z1set). This term is zero for phase-to-phase elements.        Z1set=positive sequence line replica impedance.        Z0set=zero sequence line replica impedance.        ULxpola=polarization voltage.        
FIG. 1 illustrates examples of forward directional self-polarized circular characteristics in two cases: A) a fault inside the protection zone and B) a fault outside the protection zone. Note that since S1 and S2 are voltage phasors, the characteristic is drawn in the voltage plane.
From FIG. 1 it can be seen that the angle α between S1 and S2 can be used to detect whether the measured impedance lies inside the circle. If the angle α becomes greater than 90 degrees, the measured impedance lies inside the circle and a trip signal should be generated. At the circumference the angle α is 90 degrees.
Typically the angle comparison is implemented based on a torque-like relay algorithm utilizing either a cosine or sine phase comparator. These phase comparators emulate the behavior of an induction cup element, the amplitude representing the rotating cup torque and sign rotation direction.
With the cosine-comparator the torque-like equation is:Tmho=re(S1)·re(S2)+im(S1)·im(S2)  (eq 2.3)
If Tmho<0 then impedance is inside the zone.
FIG. 1 is drawn assuming that the polarizing phasor S2 was the voltage of the faulty phase. Such polarization method is called “self polarization”. The drawback of self polarization is that in case of a close-in fault, the measured voltage may become too small and trip decision may become uncertain or delayed. Therefore, the polarizing voltage is typically chosen such as to have influence from the healthy phases. The most common types of polarization methods are cross- (or quadrature) polarization and positive sequence polarization. The polarization voltage may additionally include a memorized voltage part prior to fault inception to cope with close-in three phase faults. Table 1 shows cross- (or quadrature) polarization voltages for different fault loops.
TABLE 1FaultActual faultCross- (or quadrature)looploop voltagepolarization voltageL1EUL1j · UL23/√3L2EUL2j · UL31/√3L3EUL3j · UL12/√3L12UL12−j · UL3 · √3 or j · (UL23 − UL31)/√3L23UL23−j · UL1 · √3 or j · (UL31 − UL12)/√3L31UL31−j · UL2 · √3 or j · (UL12 − UL23)/√3
FIG. 2 illustrates voltage triangles of a symmetrical three-phase system. The positive sequence polarization voltage U1 can be calculated based on either phase-to-earth or phase-to-phase voltages as indicated in Table 2, which shows positive sequence polarization voltages for different fault loops.
TABLE 2FaultActual faultPositive sequencelooploop voltagepolarization voltageL1EUL1U1 = (UL1 + α*UL2 + α2*UL3)/3U1 = (UL12 − α2*UL23)/3U1 = (UL12 + α*UL23 + α2*UL31) ·1∠−30°3 · {square root over (3)}L2EUL2 U1 · 1∠−120°L3EUL3U1 · 1∠120°L12UL12U1 · √3 · 1∠30° L23UL23U1 · √3 · 1∠−90°L31UL31U1 · √3 · 1∠150°
An often undocumented feature of circular impedance characteristics is that the selection of polarization voltage affects the shape of the characteristics. In case of short-circuit faults the circle expands as a function of source impedance magnitude. In case of an earth fault, the circle expands as a function of source impedance and earthing impedance magnitude. The magnitude of expansion depends on the polarization method and it is different for cross-polarization and for positive sequence polarization. An example of circular characteristic expansion is illustrated in FIG. 3.
From FIG. 3 it can be noticed that the ohmic reach at angle θ is not affected by the expansion. The expansion of the characteristic is in fact a desired feature as it offers a better resistive coverage for close-in faults. Also the directionality is not affected by the expansion.
The circular characteristic is commonly used in distance protection relays. The application is typically short-circuit protection.
For earth-fault protection the desired shape of the protection zone is typically quadrilateral (polygonal), which enables detection of earth-faults with high fault resistance for the total line reach. The distance protection function with quadrilateral characteristics is typically implemented using an impedance mapping approach. This means that an impedance estimate is first calculated and the result is then compared in the impedance plane with operation zone boundaries. A typical quadrilateral characteristic has four or five boundary lines.
Typically distance protection terminals or relays include both circular and quadrilateral characteristics such that circular characteristics are implemented utilizing torque-like algorithms, while quadrilateral characteristics are implemented using the impedance mapping approach. A problem with this kind of arrangement is that the distance protection terminal or relay becomes complex.